The number of non-isomorphic arithmetic expressions that can be constructed using +,-,x and /

TL;DR

The paper tackles the problem of counting non-isomorphic arithmetic expressions (AE) formed from $n$ variables using binary operators $+$, $-$, $\times$, and $\div$, up to permutation of variables. It develops a rigorous framework based on generalized multivariate functions, a multilinear form representation, and a Fundamental Theorem that yields canonical decompositions by end-type; this enables explicit enumeration via partitions and multisets, encapsulated in Theorem 4.3. The authors provide exact counts for small $n$, derive recursive formulas via a colored-weights construction, and connect the results to the classic 21-puzzle by identifying a unique $A_4$ AE that achieves the target value. The work advances combinatorial understanding of expression structures under symmetry and delivers practical formulas and examples for the growth of $|\overline{\mathcal{A}_n}|$, with implications for rational function representations and isomorphism-aware enumeration.

Abstract

The goal of this paper is to count the number of distinct functions of n variables, up to permutation of the variables, that can be constructed using each variable exactly once, without constants, using only the operations of addition, subtraction, multiplication, and division. We refer to such a function as an arithmetic expression. Under this definition, two expressions are identical if they represent the same rational function; for example, $x_1-x_2-x_3$ and $x_1-(x_2+x_3)$ are identical arithmetic expressions, as are $x_1(x_2+x_3)$ and $(x_2+x_3)x_1$. Two arithmetic expressions are said to be isomorphic if one can be obtained from the other by a permutation of the variables. For example, $(x_1-x_2)/x_3$ and $(x_2-x_3)/x_1$ are isomorphic. The first few values of the number of non-isomorphic arithmetic expressions with n variables are: $$1,4,18,93,500,2844,16621,99674,608448,...$$ In order to accomplish this enumeration, we classify the set of all arithmetic expressions into 12 disjoint categories. Counting all non-isomorphic expressions in each category allows us to obtain the total required quantity.

Theorems & Definitions (6)

• Theorem 1: The Fundamental Theorem of Arithmetic Expressions
• Lemma 4.1: The colored weights problem
• proof
• Theorem 4.2
• Theorem 4.3
• proof